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Properties

Label 20.0.32077213915...8225.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{2}\cdot 11^{2}\cdot 47^{10}\cdot 449^{2}$
Root discriminant $18.85$
Ramified primes $5, 11, 47, 449$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 96, 8, -580, -118, 1793, 572, -2040, 2123, -1011, -1854, 2707, -504, -1002, 604, -4, -59, -23, 29, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 29*x^18 - 23*x^17 - 59*x^16 - 4*x^15 + 604*x^14 - 1002*x^13 - 504*x^12 + 2707*x^11 - 1854*x^10 - 1011*x^9 + 2123*x^8 - 2040*x^7 + 572*x^6 + 1793*x^5 - 118*x^4 - 580*x^3 + 8*x^2 + 96*x + 32)
 
gp: K = bnfinit(x^20 - 9*x^19 + 29*x^18 - 23*x^17 - 59*x^16 - 4*x^15 + 604*x^14 - 1002*x^13 - 504*x^12 + 2707*x^11 - 1854*x^10 - 1011*x^9 + 2123*x^8 - 2040*x^7 + 572*x^6 + 1793*x^5 - 118*x^4 - 580*x^3 + 8*x^2 + 96*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 29 x^{18} - 23 x^{17} - 59 x^{16} - 4 x^{15} + 604 x^{14} - 1002 x^{13} - 504 x^{12} + 2707 x^{11} - 1854 x^{10} - 1011 x^{9} + 2123 x^{8} - 2040 x^{7} + 572 x^{6} + 1793 x^{5} - 118 x^{4} - 580 x^{3} + 8 x^{2} + 96 x + 32 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32077213915073610468058225=5^{2}\cdot 11^{2}\cdot 47^{10}\cdot 449^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.85$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 11, 47, 449$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{12} a^{15} + \frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{5}{12} a^{8} - \frac{1}{3} a^{7} - \frac{5}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12840} a^{18} - \frac{31}{12840} a^{17} + \frac{403}{12840} a^{16} + \frac{659}{12840} a^{15} + \frac{853}{4280} a^{14} + \frac{301}{2140} a^{13} + \frac{151}{642} a^{12} - \frac{103}{428} a^{11} + \frac{513}{1070} a^{10} - \frac{553}{2568} a^{9} - \frac{119}{1605} a^{8} - \frac{1949}{4280} a^{7} + \frac{253}{12840} a^{6} - \frac{433}{1284} a^{5} - \frac{291}{1070} a^{4} - \frac{4423}{12840} a^{3} + \frac{357}{1070} a^{2} + \frac{24}{535} a - \frac{44}{535}$, $\frac{1}{34406586920791835952240} a^{19} - \frac{191650066735447109}{6881317384158367190448} a^{18} + \frac{205745066549644894339}{11468862306930611984080} a^{17} - \frac{1682173411170006135763}{34406586920791835952240} a^{16} + \frac{1500189015871590829513}{34406586920791835952240} a^{15} + \frac{179841046519407188777}{860164673019795898806} a^{14} - \frac{4143284336479402259}{40194610888775509290} a^{13} - \frac{547586926440729502969}{3440658692079183595224} a^{12} - \frac{401069033479320931394}{2150411682549489747015} a^{11} - \frac{1800934906178785720549}{34406586920791835952240} a^{10} - \frac{102445751996165401637}{747969280886779042440} a^{9} - \frac{14490162107594691205999}{34406586920791835952240} a^{8} + \frac{7933771593643590812011}{34406586920791835952240} a^{7} - \frac{1287537507056536746233}{8601646730197958988060} a^{6} - \frac{1252446224500051999789}{4300823365098979494030} a^{5} + \frac{1946877980258826051013}{6881317384158367190448} a^{4} - \frac{910788452817231443087}{17203293460395917976120} a^{3} - \frac{53292930577324234530}{143360778836632649801} a^{2} - \frac{359508534939066707243}{860164673019795898806} a + \frac{727112779028444224694}{2150411682549489747015}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 25174.7064629 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.229345007.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
449Data not computed